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The rule of 72 states that if I earn n% interest, I can divide 72 by n and get the approximate number of years that I'll double my money.

Suppose that I wanted to find out how many years to quadruple my money? For an example, what percent of a return would I need to generate every year in order to quadruple my money in twenty years?

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    See this answer for comparisons of how the rule of 72 works with respect to doubling. – Dilip Sarwate Jul 13 '16 at 13:06
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For 3X, it's about 114, and for 4X, 144, which naturally, is twice 72.

These are close, back of napkin, results. With smart phone apps offering scientific calculators, you should get comfortable just taking the nth root of a number for a more precise answer.

Update in response to Brick's comment.

The rule of 72 says that (n)(y)=72 to double your money. It answers both questions, how much time do I need, given a rate, and how much return do I need, given a time?

Logic tells me that if 72 is the number to double, 144 is the 4X. But I'm a math guy, and my logic might not be logical to OP. So -

Let's take the 20th root of 4.

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This is the key to use. 4, (hit key) 20, equals. The result is 1.07177 or 7.177%. And this is the precise rate you'd need to quadruple your money in 20 years) Now (n)(y)= 20* 7.177 = 143.55 which rounds to 144. "Rule of 144" to quadruple your money.

This now answers OP's question, "How to derive a Rule of X" for a return other than doubling.

One more example? I want 10X my money. Of course I need the initial guess to enter one calculation. People like 8%, in general. It's a bit below the 10% long term S&P return, and a good round number. The Rule of 72 says 9 years to double, so, 18 years is 4X, and 36 years is 8X. For my initial calculation, I'll use 40 years. The 40th root of 10. I get 5.925% (Again the precise rate that gives 10 fold over 40 years) and multiplying this by 40, I get a "Rule of 237" which I'm tempted to round to 240.

At 6%, 237/6= 39.5 yrs, 1.06^39.5 = 9.99 At 6%, 240/6= 40.0 yrs, 1.06^40.0 = 10.29

You can see that you lose some accuracy for the sake of a number that's easier to remember, and manipulate. 72 to double is pretty darn accurate, so I'll stick with "Rule of 237" to get 10X my money.

To close, the purpose of these rules is to create the tool that lets you perform some otherwise tough calculations away from any electronic device. Of course I know how to use logs, and in real life I'm paid to explain them to students who are typically glad when that chapter is over. I've shown above how the "Rule of X" can be formulated with a power/root key, which, for most people, is simpler. Ironically, log calculations as @jkuz offered, force a continuous compounding which may not be desired at all. It would give a result of 230 for my 10X return example, and the following (using the first equation he offered) -

At 6%, 230/6= 38.3 yrs, 1.06^38.3 = 9.31

which is further away from the desired 10X than my 237 or rounded 240.

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    What do you mean by nth root? Wouldn't is it be log(MULTIPLE)/log(1+RATE) = x to find how many years (X) you'd need RATE return to increase your return by a factor of MULTIPLE? Perhaps I am missing some logic. – Lan Jul 11 '16 at 19:31
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    The 10th root of 2, i.e. 2 ^ (1/10), will tell you the rate to double in 10 years. This math is a lower level than logs. – JTP - Apologise to Monica Jul 11 '16 at 19:34
  • I always have to re-derive that... – keshlam Jul 11 '16 at 19:42
  • I have an excuse. I work in a high school. And exponents/roots seem far easier to understand than the chapter introducing logs. – JTP - Apologise to Monica Jul 11 '16 at 19:59
  • The OP asked two question. The first asks for the number of years (presumably given a rate). The second asks for the rate given a number of years. For the former, he'll have to take the log, as in the comment by @Lan. For the latter, you've given the answer, but your comment about the log being harder is really misleading since it's either the right or the wrong tool depending on which of these you want to compute. – user32479 Jul 12 '16 at 13:21
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The simple answer for the so-called 'Rule of X' would be found by:

X = ln(multiple of growth) * 100 

In your case:

X = ln(4) * 100 ≈ 139

Update:

If you want an approximation closer to the nominal "Rule of 72" value, use this equation that incorporates a better approximation for the natural log. The "Rule of 72" fits this for an interest rate of 7.79% for a growth multiple of 2:

X = ln(multiple of growth) * ( 1 + ( R / 200)) * 100 

The rule of 72 comes from by approximating the natural logarithms as such:

 time =  ln(2) / ln(1+r)  ≈  0.6931/r

The 2 is the multiple of growth. The rate r here is not in percent, so to change to percent (say, R) you have to multiply by 100:

 time ≈ (0.6931 * 100) / R ≈ 69.3 / R 

The number 72 is often used because it is easier to divide evenly than 69.3 and is a better fit approximation for the natural log and common interest rates.

If you need more, you can find all this on Wikipedia:

https://en.wikipedia.org/wiki/Rule_of_72#Derivation

  • This has some degree of technical accuracy, but it misstates why 72 is used. (If it were simply because it is easier to remember than 69.3, why not use 69? or 70?) The wikipedia page explains this fairly well. – Joe Jul 11 '16 at 19:05
  • @Joe point taken! Certainly seems correct on being easier to divide than 69.3 but i will remove the other. Thanks. That is why I referenced the link! – jkuz Jul 11 '16 at 19:16
  • Rather than remove it, perhaps explain why it's actually 72. (The 69.3 is correct when r is close to 0, but ln(1+r) ~= r is only true for small r; it's 72 because that is the right answer for r=0.08 which is in the ballpark of common interest rates.) – Joe Jul 11 '16 at 19:18
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    For non-continuous compounding, 72 is more accurate. To double in 10 years, requires 7.18% interest, rule of 72 says 7.2% (damn close) but your 69.3 gives, well, 6.93% (less close). – JTP - Apologise to Monica Jul 11 '16 at 22:36
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    @JoeTaxpayer The difference between the 72 and the 69 has to do with the quality of the approximation to the log not with continuous compounding! You get 69.3 from taking the Taylor expansion around r=0 to one term. You get 72 by taking a second term for interest rates that are closer to 8% (i.e. less close to 0). – user32479 Jul 12 '16 at 15:31

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